IDEAS home Printed from https://ideas.repec.org/p/ehl/lserod/87083.html
   My bibliography  Save this paper

Rising above chaotic likelihoods

Author

Listed:
  • Du, Hailiang
  • Smith, Leonard A.

Abstract

Berliner [J. Amer. Statist. Assoc., 86 (1991), pp. 983--952] identified a number of difficulties in using the likelihood function within the Bayesian paradigm which arise both for state estimation and for parameter estimation of chaotic systems. Even when the equations of the system are given, he demonstrated “chaotic likelihood functions” both of initial conditions and of parameter values in the logistic map. Chaotic likelihood functions, while ultimately smooth, have such complicated small scale structure as to cast doubt on the possibility of identifying high likelihood states in practice. In this paper, the challenge of chaotic likelihoods is overcome by embedding the observations in a higher dimensional sequence space; this allows good state estimation with finite computational power. An importance sampling approach is introduced, where pseudo-orbit data assimilation is employed in the sequence space, first to identify relevant pseudo-orbits and then relevant trajectories. Estimates are identified with likelihoods orders of magnitude higher than those previously identified in the examples given by Berliner. The pseudo-orbit data assimilation importance sampler exploits the information both from the model dynamics and from the observations. While sampling from the relevant prior (here, the natural measure) will, of course, eventually yield an accountable sample, given the realistic computational resource this traditional approach would provide no high likelihood points at all. While one of the challenges Berliner posed is overcome, his central conclusion is supported. Chaotic likelihood functions for parameter estimation still pose a challenge; this fact helps clarify why physical scientists maintain a strong distinction between the initial condition uncertainty and parameter uncertainty.

Suggested Citation

  • Du, Hailiang & Smith, Leonard A., 2017. "Rising above chaotic likelihoods," LSE Research Online Documents on Economics 87083, London School of Economics and Political Science, LSE Library.
  • Handle: RePEc:ehl:lserod:87083
    as

    Download full text from publisher

    File URL: http://eprints.lse.ac.uk/87083/
    File Function: Open access version.
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Berliner, L. Mark & MacEachern, Steven N., 1993. "Examples of inconsistent Bayes procedures based on observations on dynamical systems," Statistics & Probability Letters, Elsevier, vol. 17(5), pages 355-360, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. James Berger & Elías Moreno & Luis Pericchi & M. Bayarri & José Bernardo & Juan Cano & Julián Horra & Jacinto Martín & David Ríos-Insúa & Bruno Betrò & A. Dasgupta & Paul Gustafson & Larry Wasserman &, 1994. "An overview of robust Bayesian analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 3(1), pages 5-124, June.

    More about this item

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:ehl:lserod:87083. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: LSERO Manager (email available below). General contact details of provider: https://edirc.repec.org/data/lsepsuk.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.